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A cell partitioning of a closed 3 3 -manifold M 3 {M^3} is a finite covering of M 3 {M^3} by 3 3 cells that fit together in a bricklike pattern. A cell partitioning H H of M 3 {M^3} is shellable if H H has a counting ⟨ h 1 , h 2 , … , h n ⟩ \langle {h_1},{h_2}, \ldots ,{h_n}\rangle such that if 1 ⩽ i > n , h 1 ∪ h 2 ∪ ⋯ ∪ h i 1 \leqslant i > n,\;{h_1} \cup {h_2} \cup \cdots \cup {h_i} is a 3 3 -cell. The main result of this paper is a relationship between nonshellability of a cell partitioning H H of S 3 {S^3} and the existence of links in S 3 {S^3} specially related to H H . This result is used to construct a nonshellable cell partitioning of S 3 {S^3} .

A knotted dogbone space is a dogbone space in which each handle of each solid double tori used in defining the decomposition is nontrivially knotted. We prove that for some points p of the resulting decomposition space, p does not have arbitrarily small simply connected open neighborhoods.

F. B. Jones [2] recently gave an example of a Moore space A,, in which there exists a point x such that A,, is not completely regular at x. It is easy to modify the construction used by Jones so as to obtain a Moore space A in which there exist distinct points a and b such that for every real-valued continuous function f on A, f(a) =f(b). Upon applying Urysohn's process of condensation of the singularities of the space A [4], in a manner similar to that used by Hewitt [1], there results a Moore space X on which every real-valued continuous function is constant. Throughout this paper, J denotes the set of positive integers. A sequence is a function on J, and if f is a sequence and n E J, then f, denotes f(n) . By a Moore space is meant a topological space X whose topology has a basis consisting of sets termed regions, satisfying the following condition (axiom 13, that is, parts 1, 2, and 3 of axiom 1, of [3]): There exists a sequence G such that (1) if nE J, G. is a collection of regions covering X, (2) if nCJ, Gnu?1CG., and (3) if r is a region, xer, and yEr, then there exists a positive integer n such that if g CE Gn and x eg, then g C (rx } ) J{ y }. The following characterization of a Moore space will be used in this paper: X is a Moore space if and only if X is a regular Hausdorff space for which there exists a sequence G of open coverings of X such that if U is an open set and

1. Introduction. In [2], we proved that if G is a cellular decomposition of a 3-manifold M such that the associated decomposition space is a 3-manifold N, then M and N are homeomorphic. In this paper we shall establish a related result for 3-manifolds with boundary. We shall show that if G is a cellular decomposition of a 3-manifold with boundary M such that the associated decomposition space is a 3-manifold with boundary N, then M and N are homeomorphic. The techniques of this paper have applications to the study of embeddings of curves and surfaces in 3-manifolds. Suppose that M i's a 3-manifold with boundary and G is a cellular decomposition of M such that the7 associated decomposition space is a 3-manifold with boundary N. By Theorem 2 of this paper, M anld N are homeomorphic. Suppose K is a surface or a curve in M such that no nondegenerate element of G intersects K. If P denotes the projection map from M onto N, then PIK is a homeomorphism. It is natural to ask the following: Is P[K]'embedded in N the same way that K is embedded in M? In ?5, we give an affirmative answer to this question. In particular, K is tame if and only if P[KJ is tame. In ?6 we shall show that if G is a cellular decomposition of a 3-manifold with boundary into a 3-manifold with boundary, then, the projection' map can be approximated arbitrarily closely by homeomorphisms. In [2]; we established a theorem of basic importance in the study of cellular decompositions of 3-manifoids that yield 3-manifolds as their decomposition spaces. The main result, Theorem 1, of this paper is a useful corollary of the results of [2]. The results mentioned in the preceding two paragraphs are applications of Theorem 1. In [4], we shall apply Theorem 1 to the study of shrinkability conditions which are satisfied by certain cellular decompositio'ns of E3 that yield El as their decomposition space. 2. Terminology and notation. The statement that M is a 3-manifold with boundary means that M is a separable metric space such that each point of M has a neighborhood in M which is a 3-cell. If M is a 3-manifold with boundary, a point p of M is an interior point of M if and only if p has an open neighborhood in M which is an open 3-cell. The interior of M, mnt M, is the set of all boundary points

In this paper, certain separation theorems, which hold in spaces satisfying R. L. Moore's Axioms 0 and 1-5 of [2], are established for a class of spaces in which there do not necessarily exist either arcs or simple closed curves. The spaces considered satisfy axioms suggested by those of Moore. Consider a nondegenerate complete metric space S which satisfies the following axioms from [2]: AxioM 2. S is locally connected. AXIoM 3. S is connected and has no cut point. AXIoM 4. If J is a simple closed curve in S, then S J has exactly two components and J is the boundary of each of them. AXIoM 5. If x and y are distinct points and U is an open set containing x, then there is a simple closed curve J in U such that J separates x from y. It will be shown that even with a weakening of these conditions on S, the following separation theorems hold: (1) If neither of the compact point sets H and K separates the two points x and y, and either H and K are disjoint or HOK is connected, then HkJK does not separate x and y. (2) If the common part of the two compact continua H and K is not connected, then HUK separates S. Principally, the conditions above are weakened by the omission of the stipulation that S be complete. It is assumed instead that S is locally compactly connected (a point set K in S is compactly connected if and only if each two points of K belong to a compact continuum in K). As an example in ?4 shows, in a locally compactly connected metric space, there does not necessarily exist an arc. However, there exist chains of compact continua, and sets which are the unions of links of such chains serve as substitutes for arcs. The sets which serve as substitutes for simple closed curves are the unions of the links of closed chains of compact continua. Moore's Axioms 4 and 5 are replaced by analogous axioms for closed chains. This problem was suggested to the writer by R. L. Moore, and the development presented here is along lines similar to that in [2] which leads to the separation theorems mentioned above. 1. Consequences of Axioms 1 and 2. AXIOM 1. S is a nondegenerate connected metric space with no cut point. The diameter of a bounded point set M and the distance between two point sets H and K are denoted by diam M and dist (H, K), respectively.